General failure of Rieffel’s Condition 1.5 for reduced groupoid C*-algebras

The paper constructs Lipschitz quasi-seminorms L_ℓ^k on reduced groupoid C*-algebras C_r^*(𝔾) arising from continuous length functions on étale groupoids and studies the induced pseudo-metrics on the state space. Because ker(L_ℓ^k) contains C(X) for transformation groupoids 𝔾 = Γ ⋉ X, the metric on the entire state space is a pseudo-metric and one must restrict attention to fibres S_η determined by probability measures η on X.

For transformation groupoids with a continuous, proper length function having rapid decay, the authors show each fibre (S_η, ρ{Lℓ^k}) has uniformly bounded diameter, and when X is finite, ρ{Lℓ^k} metrizes the weak*-topology, yielding a quasi-compact quantum metric space. However, establishing necessity of the total boundedness condition (used to show metrization) typically requires an inequality like Rieffel’s Condition 1.5 to pass from function-space estimates to the C*-algebra setting. The authors conjecture that this inequality fails in general for reduced groupoid C*-algebras, even for transformation groupoids, leaving open the precise status of that inequality in the groupoid context.